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AMERICAN MATHEMATICAL SOCIETY
Volume 363, Number 9, September 2011, Pages 4701–4733
S 0002-9947(2011)05197-4
Article electronically published on March 28, 2011
CUBULATING RANDOM GROUPS
AT DENSITY LESS THAN 1/6
YANN OLLIVIER AND DANIEL T. WISE
Abstract. We prove that random groups at density less than 16 act freely and
cocompactly on CAT(0) cube complexes, and that random groups at density
less than 15 have codimension-1 subgroups. In particular, Property (T ) fails
to hold at density less than 15 .
Abstract. Nous prouvons que les groupes aléatoires en densité strictement
inférieure à 16 agissent librement et cocompactement sur un complexe cubique
CAT(0). De plus en densité strictement inférieure à 15 , ils ont un sous-groupe
de codimension 1; en particulier, la propriété (T ) n’est pas vérifiée.
Introduction
Gromov introduced in [Gro93] the notion of a random finitely presented group
on m 2 generators at density d ∈ (0; 1). The idea is to fix a set {g1 , . . . , gm } of
generators and to consider presentations with (2m − 1)d relations each of which
is a random reduced word of length (Definition 1.1). The density d is a measure
of the size of the number of relations as compared to the total number of available
relations. See Section 1 for precise definitions and basic properties, and see [Oll05b,
Gro93, Ghy04, Oll04] for a general discussion on random groups and the density
model.
One of the striking facts Gromov proved is that a random finitely presented
group is infinite, hyperbolic at density < 12 , and is trivial or {±1} at density > 12 ,
with probability tending to 1 as → ∞.
Żuk obtained Property (T ) for a related class of presentations at density > 13
(see [Żuk03] and the discussion in [Oll05b]). On the other hand, Gromov observed
that at density < d, a random presentation satisfies the C (2d) small cancellation
1
, the groups will not have Property (T )
condition. Consequently, at density < 12
1
since C ( 6 ) groups act properly discontinuously on CAT(0) cube complexes [Wis04].
As above, the statements about the behavior of a group at a certain density are
only correct with probability tending to 1 as → ∞. Throughout the paper, we
will say that a given property holds with overwhelming probability if its probability
tends exponentially to 1 as → ∞.
Received by the editors August 27, 2008 and, in revised form, August 27, 2009.
2010 Mathematics Subject Classification. Primary 20P05, 20F67; Secondary 20F65, 20F05,
20F06.
Key words and phrases. CAT(0) cube complexes, random groups, Property (T ).
This research was partially supported by NSERC grant.
c
2011
American Mathematical Society
Reverts to public domain 28 years from publication
4701
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Y. OLLIVIER AND D. T. WISE
The goals of this paper are a complete geometrization theorem at d < 16 , implying the Haagerup property, and existence of a codimension-1 subgroup at d < 15 ,
implying [NR98] failure of Property (T ):
Theorem 10.4. With overwhelming probability, random groups at density d <
act freely and cocompactly on a CAT(0) cube complex.
1
6
Corollary 9.2. With overwhelming probability, random groups at density d <
are a-T-menable (Haagerup property).
1
6
Theorem 7.4. With overwhelming probability, random groups G at density d < 15
have a subgroup H which is free, quasiconvex and such that the relative number of
ends e(G, H) is at least 2.
Corollary 7.5. With overwhelming probability, random groups at density d <
do not have Property (T ).
1
5
CAT(0) cube complexes are a higher dimensional generalization of trees, which
arise naturally in the splitting theory of groups with codimension-1 subgroups
[Sag95, Sag97]. A group is a-T-menable or has the Haagerup property [CCJ+ 01]
if it admits a proper isometric action on a Hilbert space. This property is, in a
certain sense, an opposite to Kazhdan’s Property (T ) [dlHV89, BdlHV08], which
(for second countable, locally compact groups) is characterized by the requirement
that every isometric action on an affine Hilbert space has a fixed point. There is
also a definition of the Haagerup property in terms of a proper action on a space
with measured walls [CMV04, CDH], which is a natural framework for some of our
results.
The relative number of ends e(G, H) of the subgroup H of the finitely generated
group G is the number of ends of the Schreier coset graph H\G (see [Hou74, Sco78]).
Note that e(G, H) is independent of the choice of a finite generating set. We say
that H is a codimension-1 subgroup of G if H coarsely disconnects the Cayley graph
Γ of G, in the sense that the complement Γ − Nk (H) of some neighborhood of H
contains at least two components that are not contained in any finite neighborhood
Nj (H) of H. The above two notions are very closely related and are sometimes
confused in the literature: If e(G, H) > 1, then H is a codimension-1 subgroup of
G, and the converse holds when there is more than one H-orbit of an “infinitely
deep” component in Γ − Nk (H).
Let us present the structure of the argument. In [Sag95], Sageev gave a fundamental construction which, from a codimension-1 subgroup H of G, produces an
“essential” action of G on a CAT(0) cube complex. From [NR97] or [NR98] we
know, in turn, that groups acting essentially/properly on a CAT(0) cube complex
act essentially/properly on a Hilbert space and cannot have Property (T ) (their
proof is a generalization of a proof in [BJS88] that infinite Coxeter groups are a-Tmenable, which in turn was a generalization of Serre’s argument that an essential
action on a tree determines an essential action on a Hilbert space [Ser80]).
In our situation the codimension-1 subgroups will arise as stabilizers of some
of the
codimension-1 subspaces, called hypergraphs, in the Cayley 2-complex X
random group G. These hypergraphs are the same as those in [Wis04] and are
defined in Section 2. The basic idea is, from the midpoint of each 1-cell in a 2-cell
c, to draw a line to the midpoint of the opposite 1-edge in c (assuming all 2-cells
have even boundary length). These lines draw a graph in the 2-complex, whose
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
4703
connected components are the hypergraphs. Hypergraphs are natural candidates
to be walls [HP98].
In Section 4 we show that at density d < 15 , with overwhelming probability, the
hypergraphs embed (quasi-isometrically) in the Cayley 2-complex. The main idea
is that if a hypergraph self-intersects, it will circle around a disc in the Cayley
2-complex, thus producing a collared diagram (Section 3). But at d < 15 , the
Dehn algorithm holds for a random group presentation [Oll07], so that in each van
Kampen diagram some 2-cell has more than half its length on the boundary, which
is impossible if a hypergraph runs around the boundary 2-cells of the diagram.
A consequence of this embedding property is that each hypergraph is a tree
into two connected components, thus turning X
into a space with
dividing X
walls [HP98].
We then show that these walls can be used to define (free quasiconvex) codimension-1 subgroups (Section 7). For this we need the complex X̃ to go “infinitely far
away” on the two sides of a given wall. This is guaranteed by exhibiting a pair of
infinite hypergraphs intersecting at only one point. At d < 16 , hypergraphs intersect
at at most one point except for a degenerate case (Section 5). This is not true in
general for d < 15 ; however, one can still prove that through a “typical” 2-cell that
a hypergraph Λ1 passes through, there passes a second hypergraph Λ2 transverse
to Λ1 , which is enough (Section 6).
To prove the Haagerup property, we show that at d < 16 , the number of hypergraphs separating given points p, q ∈ X̃ is at least distX̃ (p, q)/K for some constant
K. Consequently the wall metric is quasi-isometric to the Cayley graph metric,
which implies that the group has the Haagerup property. Key objects here are hypergraph carriers (the set of 2-cells through which a hypergraph travels): at d < 16
these carriers are convex subcomplexes of X̃, but this is not the case at d > 16 . We
were unable to prove that points are separated by a linear number of hypergraphs
at 16 d < 15 , where the failure of convexity substantially complicates matters,
though we conjecture that such a statement still holds.
Finally, Theorem 10.4 is proven by combining the various properties established
at d < 16 (including, most importantly, the separation of points by a linear number
of hypergraphs) to see that the cubulation criteria in [HW04] are satisfied; these
criteria guarantee that the action of G on the CAT(0) cube complex associated with
a codimension-1 subgroup arising from hypergraphs is indeed free and cocompact.
At density d > 15 , our approach completely fails: with overwhelming probability,
there is only one hypergraph Λ, which passes through every 1-cell of the Cayley
complex (Section 11). Its stabilizer is the entire group, and it is thus certainly not
codimension-1. We do not know if there are codimension-1 subgroups at density
1
1
1
5 < d < 3 . But, as mentioned above, the transition at d = 5 in the behavior
of hypergraphs is related to another one, namely failure of the Dehn algorithm
for d > 15 [Oll07], and our intuition is that something of both combinatorial and
geometric relevance really happens at d = 15 .
1. Preliminaries and facts regarding Gromov’s density
The density model of random groups was introduced by Gromov in [Gro93],
Chapter 9 as a way to study properties of “typical” groups depending on the quantity of relators in a presentation of the group. We refer to [Oll05b, Gro93, Ghy04,
Oll04] for general discussions on random groups and the density model.
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Y. OLLIVIER AND D. T. WISE
Definition 1.1 (Density model of random groups). Let m 2 be an integer and
±1
consider the free group Fm generated by a±1
1 , . . . , am .
Let 0 d 1 be a density parameter. Let be a (large) length. Choose
(2m − 1)d times (rounded to the nearest integer) at random a reduced word of
±1
length in the letters a±1
1 , . . . , am , uniformly among all such words. Let R be the
set of words so obtained.
A random group at density d and length is the group G = Fm /R, whose
presentation is a1 , . . . , am | R .
A property is said to occur with overwhelming probability in this model if its
probability of occurrence tends exponentially to 1 as → ∞.
Note that, a priori, repetitions are allowed in the choice of the random words (so
that the choices are independent); but actually when d < 1/2, with overwhelming
probability there are no repetitions.
The basic intuition is that at density d, subwords of length (d−ε) of the relators
in the presentation will exhaust all reduced words of length (d − ε).
The interest of the model is established through the following sharp phase transition theorem, proven by Gromov [Gro93] (see also [Oll04]):
Theorem 1.2 (M. Gromov). Let d < 1/2. Then with overwhelming probability, a
random group at density d is infinite, hyperbolic, and torsion-free.
Let d > 1/2. Then with overwhelming probability, a random group at density d
is either {1} or {1, −1}.
One of the motivations for the results in this paper is the following ([Żuk03]; see
also the discussion in [Oll05b]):
Theorem 1.3 (A. Żuk). Let d > 1/3. Then with overwhelming probability, a
random group at density d has Property (T ).
It is not known whether 1/3 is optimal in this theorem. Our results imply that
1/5 is a lower bound.
Remark 1.4. According to the definition above, all relators in a random group have
exactly the same length. However, the results stay the same if we take relators of
length between and + C, where C is any constant independent of .
Some results on random groups, including Theorem 1.2 and Theorem 1.3, also
extend to the case when relators are taken of length between and C for some
C > 1 (see [Oll04]), but we do not know if this is the case for the main theorems
presented in this paper.
Hyperbolicity of random groups at d < 1/2 is proven using isoperimetry of van
Kampen diagrams. In this paper we shall repeatedly need a precise statement of
this isoperimetric inequality, which we now state.
For a van Kampen diagram D, we use the notation |∂D| for the length of its
boundary path, and the notation |D| for the number of 2-cells in D.
Convention 1.5. When a property of a random group depends on a parameter
ε, the phrase “the property occurs with overwhelming probability” will mean that
for any ε > 0, the probability of the property tends exponentially to 1 as → ∞.
(This may not be uniform in ε.)
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
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The following, proven in [Oll07], is a strengthening of the original statement of
Gromov, which held only for diagrams of size bounded by some constant. Note the
role of d = 12 .
Theorem 1.6. At density d, for any ε > 0 the following property occurs with
overwhelming probability: all reduced van Kampen diagrams D satisfy
|∂D| (1 − 2d − ε) |D| .
When using this result in this paper we will often omit the ε.
We now gather some definitions pertaining to small cancellation. We refer to
chapter V of [LS77] for the definition of a piece in a group presentation.
Definition 1.7 (Small cancellation). A presentation satisfies the C (α) condition,
with 0 α 1, if for each relator R, and each piece P occurring in R, we have
|P | < α |R|.
A presentation satisfies the B(2p) condition if every word w which is a concatenation of at most p pieces and which is a subword of a relator R satisfies |w| 12 |R|.
A presentation satisfies the C(p) condition if no relator R is the concatenation
of fewer than p pieces.
1
) ⇒ B(2p) ⇒ C(2p) but that none of the reverse implications
Note that C ( 2p
holds.
Proposition 1.8. With overwhelming probability:
(1) The C (α) condition occurs at density < α/2.
(2) The B(6) condition occurs at density < 18 .
(3) The C(p) condition occurs at density < p1 .
Proof. The proof for C (α) is written in detail in [Gro93], §9.B. Let us briefly recall
the argument. Since the number of reduced words of length L is (2m)(2m − 1)L−1 ,
the probability that two random reduced words of length share a common initial
−1
subword of length L is (2m)(2m − 1)L−1
(2m − 1)−L .
So given two random words of length , the probability that they share a piece
of length L is less than 2 (2m − 1)−L , where the 2 accounts for the choice of the
position at which the piece occurs.
Now in a random group at density d, there are by definition (2m − 1)d relators.
So the probability that there exists a couple of relators in the presentation having
a piece of length L is at most 2 (2m − 1)2d (2m − 1)−L since there are (2m − 1)2d
possible choices of couples of relators (we also have to check the special case when
a relator shares a piece with itself, but this is not difficult). So if L = α this makes
2 (2m − 1)(2d−α) . If d < α/2, this tends to 0 as → ∞ (but all the more slowly
as d is close to α/2). One can reverse the argument to see that if d > α/2, such an
event actually occurs.
It is worthwhile to compare this with Theorem 1.6. Indeed, when two relators
share a piece of length α we can form a van Kampen diagram D of boundary
length |∂D| = 2 − 2α = |D| (1 − α) so that this diagram contradicts Theorem 1.6
when d < α/2.
The C(p) condition amounts to the exclusion of a reduced van Kampen diagram
D in which a 2-cell is surrounded by at most p − 1 2-cells as on the left of Figure 1.
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4706
Y. OLLIVIER AND D. T. WISE
Figure 1. Diagrams contradicting the C(7) and B(6) conditions.
p(1 − 2d − ε) so that (choosing ε = (1/p − d)/10) this is a contradiction when
d < 1/p. This proves statement (3).
The B(6) condition amounts to the exclusion of a diagram in which half the
boundary of a 2-cell is covered by three other 2-cells as on the right of Figure 1.
Note that this diagram D satisfies |∂D| = 4 − 2(/2) = 3. Theorem 1.6 implies
that |∂D| 4(1 − 2d − ε), so d 1/8 − ε/2. So if d < 1/8 we get a contradiction
(choosing e.g. ε = (1/8 − d)/10).
Remark 1.9. By [Wis04], hyperbolicity and the B(6) condition together imply the
existence of a free and cocompact action on a CAT(0) cube complex. So this
1
condition
conclusion holds at density < 18 . This is a bit stronger than the < 12
mentioned in the introduction.
Since the C(6) condition is satisfied at density < 16 , our results suggest that
generic C(6) groups are a-T-menable. It is currently an open problem whether or
not every infinite C(6) group fails to satisfy Property (T ).
In this paper we shall sometimes need to avoid some annoying topological configuration. This is the object of the next two propositions.
Proposition 1.10. Let G be a random group at density d < 1/4. Let p be a closed
path embedded in the Cayley graph of G. Then the length of p is at least ; moreover,
either p is the boundary path of some relator in the presentation, or the length of p
is at least + (1 − 4d − ε).
Consequently, the boundary paths of relators embed.
Proof. This results from Theorem 1.6. Indeed, since p is not homotopic to 0, it is
the boundary path of some van Kampen diagram D with at least one 2-cell, and
so |p| |D| (1 − 2d − ε). Now either |D| = 1 and p is the boundary path of a
relator, or |D| 2 and |p| 2(1 − 2d − ε).
Corollary 1.11. Let G be a random group at density d < 1/4 and let
Cayley complex associated to the presentation. Let c1 , c2 be two 2-cells in
∂c1 ∩ ∂c2 is connected.
be the
X
Then
X.
lying in different components
Proof. Suppose not and let v, w be two 0-cells of X
of ∂c1 ∩ ∂c2 . Let p1 , p1 be the two paths in ∂c1 joining v to w on each side of c1 ,
and likewise let p2 , p2 be the two paths in ∂c2 joining v to w.
−1
−1 −1
, p1 p2 and p1 p2
is a closed path in the 1Each of the paths p1 p−1
2 , p1 p2
skeleton of X̃. Each of these paths is not null-homotopic in this 1-skeleton; otherwise v and w would lie in the same component of ∂c1 ∩ ∂c2 . So by Proposition 1.10
each of these paths has length at least , and since |p1 | + |p1 | = |p2 | + |p2 | = ,
the only
possibility is that |p1 | = |p1 | = |p2 | = |p2 | = /2. This implies that
p1 p−1 = , so that p1 p−1 is the boundary path of some 2-cell c3 . Now c1 and
2
2
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
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c3 share half of their boundary length, which at d < 1/4 contradicts Proposition 1.8.
Another notion we shall need is that of a fulfilling of a diagram. Let D, an
abstract diagram, be a finite connected graph embedded in the plane, each edge of
which is decorated with a positive integer, its length. Let a1 , . . . , am | R be any
group presentation. A fulfilling of D is the attribution to each face of D of a relator
in R (together with an orientation) such that the resulting object is a reduced
van Kampen diagram of the presentation, in a way compatible with the prescribed
lengths (see the notion of decorated abstract van Kampen diagram in [Oll04] for
details).
The following appears in [Oll05a], Propositions 12 and 13:
Theorem 1.12. Let G = a1 , . . . , am | R be a group presentation. For any
abstract diagram D, let Sn (D) be the number of n-tuples of distinct relators in R
such that there exists a fulfilling of D using these relators (n is at most the number
of faces |D| of D since a relator may be used multiple times in the diagram).
For random groups at density d, for any abstract diagram D we have the following
bound on the expectation of Sn (D):
1
ESn (D) (2m − 1) 2 (|∂D|−(1−2d)|D|)
and so for any D, with overwhelming probability we have
1
Sn (D) (2m − 1) 2 (|∂D|−(1−2d−ε)|D|) .
We note that the second assertion in Theorem 1.12 (which holds for fixed D)
follows from the first one by the Markov inequality.
2. Hypergraphs and carriers
2.1. Historical background on cubulating groups. The results in this paper
employ Sageev’s construction [Sag95] of an action on a CAT(0) cube complex from
a group G and a codimension-1 subgroup H.
Niblo and Reeves [NR03] and Wise [Wis04] had observed that Sageev’s construction works in the context of “geometric spaces with walls”. For Coxeter groups,
these walls are the reflection walls stabilized by the involutions in the Coxeter complex. For small cancellation groups, the walls are constructed as we do here: by
producing immersed graphs in a 2-complex that are transverse to the 1-skeleton and
such that each edge of the graph bisects a 2-cell. The walls corresponding to such
graphs appear to have played a role in Ballmann-Światkowski’s proof of the failure of Property (T) for the geometric case of (4, 4)-complexes and (6, 3)-complexes
[BŚ97].
It is clear from [Wis04] that Sageev’s cubulation result can be carried out for a
family of more general codimension-1 graphs which embed, are transverse to the
1-skeleton, and locally separate the 2-complex. These are examples of Dunwoody’s
“tracks” and we expect they will be referred to as “walls” in future work on this
subject. Indeed, subsequently, Nica [Nic04] and Chatterji and Niblo [CN05] have
written out an explicit application of Sageev’s construction to cubulate abstract
“spaces with walls”. Those were introduced by Haglund and Paulin [HP98] especially motivated by Coxeter groups and CAT(0) cube complexes.
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4708
Y. OLLIVIER AND D. T. WISE
Building upon [Sag97, NR03, Wis04], Hruska and Wise [HW04] have laid out
“axioms” on a space with walls (or 2-complex with hypergraphs) for verifying finiteness properties of the cubulation. We follow the framework there to verify our main
results. We expect there will be further work along these lines.
2.2. Definition of hypergraphs. In a nutshell, hypergraphs in a 2-complex are
obtained by drawing a segment between the midpoints of each pair of opposite 1cells in each 2-cell. These segments define a graph, the connected components of
which are the hypergraphs. We give a more precise definition below.
be a simply connected 2-complex. We suppose that each
Definition 2.1. Let X
2-cell of X has even boundary length. (If this is not the case, we just perform a
before constructing hypergraphs.)
subdivision of all 1-edges of X
We define a graph Γ as follows: The set of vertices of Γ is the set of 1-cells of
There is an edge in Γ between two vertices if there is some 2-cell R of X
such
X.
that these vertices correspond to antipodal 1-cells in the boundary of R (if there
are several such 2-cells R, we put as many edges in Γ). The 2-cell R is the 2-cell of
containing the edge.
X
which
There is a natural map ϕ from Γ to a geometric realization of it in X,
sends each vertex of Γ to the midpoint of the corresponding 1-cell of X, and each
edge of Γ to a segment joining two antipodal points in the 2-cell R. Note that
the images of two edges contained in the same 2-cell R always intersect, so that in
general ϕ is not an embedding.
is a connected component of Γ. The 1-cells of X
through
A hypergraph in X
which a hypergraph passes are dual to it. The hypergraph Λ embeds if ϕ is an
i.e., if no two distinct edges of
embedding from Λ to its geometric realization in X,
Λ live in the same 2-cell of X.
we define a 2-complex V , the unfolded carrier of A,
For each subgraph A ⊂ Γ,
consider
in the following way: For each edge e in A contained in the 2-cell R of X,
an isomorphic copy Re of R. Now take the disjoint union of these copies and glue
them as follows: if edges e and e of A share a common endpoint v ∈ A, identify
Re and Re along the 1-cell corresponding to vertex v. When A is connected, it is,
by construction, an embedded hypergraph of its unfolded carrier.
A hypergraph segment (resp. ray, resp. line) is an immersed finite path (resp.
immersed ray, immersed line) in a hypergraph. A ladder is the unfolded carrier of
a segment.
Remark 2.2. The term “hypergraph” is a misnomer, which arose as a graph corresponding to a “hyperplane” in a CAT(0) cube complex C. Hypergraphs will play
the role of “codimension-1 subgraphs” later in the paper. The term hypergraph is
used in graph theory to mean a certain high-dimensional generalization of a graph,
but we will have no use for that notion in this paper.
Lemma 2.3. Suppose a hypergraph Λ is an embedded tree in the simply connected
Then X
− Λ consists of two components.
complex X.
= 0 and a Mayer-Vietoris seProof. This follows easily from the fact that H1 (X)
quence argument applied to the complement of the hypergraph and a neighborhood
of the hypergraph.
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
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3. Studying hypergraphs with collared diagrams
In this section we define and examine various notions of “collared diagrams”. In
Section 3.1, we show that hypergraphs are trees unless certain collared diagrams
exist. In Section 3.3, we show that the intersection of a pair of hypergraphs contains
at most one point, unless there is a certain collared diagram between them. In
Section 3.4, we explain that if a geodesic touches a hypergraph in exactly two
points, then there is a certain relatively collared diagram between the geodesic and
the hypergraph. In each case, various quasicollared diagrams will serve as a useful
technical object to facilitate the proofs.
and its hypergraphs). In the remainder of the
Convention 3.1 (Conventions on X
is the Cayley 2-complex of a random group (and hence all relations have
paper, X
the same length). However in this section we work under more general hypotheses
is that it is
(which the reader is welcome to ignore). Our only hypothesis on X
a simply connected combinatorial 2-complex, and that the boundary cycle of each
1 , of even length.
2-cell is an immersed path in X
3.1. Collared diagrams. We refer to [MW02] (Def. 2.6) for the definition of disc
diagrams, which play for arbitrary 2-complexes the role of van Kampen diagrams
for Cayley complexes. The reader may just read “van Kampen diagram”.
The central notion in this section is the following (see Figure 2):
Definition 3.2 (Collared diagram). A collared diagram is a disc diagram D → X
with the following properties:
(1) there is an external 2-cell C called a corner of D;
of length at least 2;
(2) there is a hypergraph segment λ → D → X
(3) the first and last edge of λ lie in C, and no other edge lies in C;
(4) λ passes through every other external 2-cell of D exactly once;
(5) λ does not pass through any internal 2-cell of D.
D is cornerless if moreover the first and last edge of λ coincide in C (in which
case the hypergraph cycles).
The above definition implies that the diagram is homeomorphic to a disc.
Remark 3.3. Note that we do not exclude that C is the only boundary 2-cell of D
(in which case the boundary path of C is not simple). However, since the boundary
path of any 2-cell is immersed by assumption, D has at least two 2-cells. But it
might not have any internal 2-cells.
Definition 3.4. A cancellable pair in Y → X is a pair of distinct 2-cells R1 , R2
meeting along an edge e in Y such that R1 and R2 map to the same 2-cell in X,
and moreover, the boundary paths of R1 and R2 starting at e map to the same
path in X. A map Y → X is reduced if Y contains no cancellable pairs. Note that
the composition of reduced maps is reduced.
In our framework, Y → X is reduced precisely if Y → X is a near-immersion
meaning that (Y − Y 0 ) → X is an immersion. See [MW02] for more about reduced
maps.
For van Kampen diagrams this notion coincides with the usual notion of reduced
diagram (at least if relators which are proper powers are handled correctly, which
is a messy point in the van Kampen diagram literature).
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4710
Y. OLLIVIER AND D. T. WISE
Figure 2. Several kinds of collared diagrams: The corner 2-cell of
the first collared diagram is shaded. The second collared diagram is
cornerless. The hypergraph segment of the third collared diagram
ends before it enters the interior. The last collared diagram is more
typical in the sense that the carrier of the segment folds. (Note
that the figure is topological only, so that some 1-cells may appear
longer or shorter than others, yet hypergraphs still cross 2-cells at
antipodal pairs of 1-cells.)
The main goal of this section is to prove the following theorem.
Theorem 3.5. Let Λ be some hypergraph. The following are equivalent:
(1) Λ is an embedded tree.
(2) There is no reduced collared diagram collared by a segment of Λ.
(3) There is no quasicollared diagram collared by a segment of Λ (Definition 3.6
below).
collared by a segment λ of Λ, then
Proof. If there is a reduced diagram E → X
clearly either Λ is not a tree or Λ → X is not an embedding. Indeed, the path
λ → E has the property that its first and last edges cross or coincide, and so this
is the case for λ → X.
The converse, which plays an important role in this paper, needs a bit more
work, and so we outline the proof which employs several lemmas proven later in
this section. Suppose Λ is not an embedded tree.
then there exists
In Lemma 3.8, we prove that if Λ is not an embedded tree in X,
a diagram quasicollared by Λ (Definition 3.6), denoted by F → X.
In Lemma 3.9, we show that by removing cancellable pairs, we can assume that
is reduced.
F →X
from the reduced
In Lemma 3.10, we extract a reduced collared diagram E → X
quasicollared diagram F → X.
We now define quasicollared diagrams which, unlike collared diagrams, do not
have an easily stated intrinsic definition.
Definition 3.6 (Quasicollared diagram). Consider the ladder L of some hypergraph segment λ of length at least 2. We suppose that the first and last 2-cells
C1 , C2 of L map to the same two-cell of X.
Let A = L/{C1= C2 } be the complex obtained from L by identifying the closures
C2 → X).
of C1 and C2 (using the maps C1 → X,
Let P → A be a simple cycle in A representing a generator of H1 (A). Suppose
with boundary path P .
that there exists a disc diagram D → X
A quasicollared diagram F → X is the complex obtained by forming the union
F = A ∪P D. (See Figure 3.) We say that F is collared by the hypergraph segment
λ.
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
4711
Figure 3. On the left is the path P in the ladder L containing
part of the hypergraph Λ. On the right is the quasicollared disc
diagram L ∪P D obtained by attaching L to D along P .
Figure 4. The basic loops.
Remark 3.7. F is a genuine disc diagram precisely when P → A does not cross λ.
This happens precisely when A is a cylinder instead of a Moebius strip and P → A
is a boundary cycle of A.
Lemma 3.8 (Existence). Suppose that the hypergraph Λ is not an embedded tree
Then there exists a quasicollared diagram F → X
that is collared by Λ.
in X.
Proof. The hypergraph Λ is not an embedded tree if and only if there exists a
nontrivial immersed edge-path λ → Λ such that λ projects to a nonsimple path in
We can assume that the hypergraph segment λ is minimal, that is, any proper
X.
subsegment of λ embeds.
containing the first and last edge of λ are the same. Let L be
So the 2-cells of X
the ladder carrying λ; its first and last 2-cells C1 , C2 map to the same 2-cell of X.
We can therefore form a quotient space A = L/{C1= C2 } and there is an induced
We will refer to the cell C1 = C2 as the corner. As in Figure 4, there
map A → X.
are two cases for A according to whether or not λ “preserves orientation” of A.
Let P → A be a simple cycle in A that maps to a generator of π1 (A). Since X
is simply connected, there is a disc diagram D → X whose boundary path is P .
may not be, and so it
Note that while P → A is an immersion, the map A → X
is possible that D is singular and may have spurs.
Finally we form the desired quasicollared diagram F = A ∪P D.
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4712
Y. OLLIVIER AND D. T. WISE
λ
λ
P
P
D
D
P
D
Figure 5. A cancellable pair between D and A, another cancellable pair in a slightly different position, and a cancellable pair
between A and A. The cancellable pairs are marked with dots.
If we think of P as a path in L instead of A, then P may travel from one side
of L to the other, as in Figure 3. Indeed, this is always the case in the orientationreversing case where A is a Moebius strip. F is a genuine disc diagram exactly
when λ does not cross any edge of P .
Lemma 3.9 (Reducing). Let F → X be a quasicollared diagram. Then there exists
a reduced quasicollared diagram F → X which is collared by a subsegment of the
hypergraph segment collaring F .
Proof. Keeping the notation in the definition of quasicollared diagrams, there are
to consider according to whether the
three types of cancellable pairs in F → X
2-cells lie in: D, D or D, A, or A, A.
In the first case, the cancellable pair is removed in the usual way for van Kampen
diagrams (prone to errors in the literature, but works nevertheless...): We remove
the open 2-cells and the open 1-cell along which they form a cancellable pair, and
we identify their remaining corresponding boundaries.
In the second case, we can adjust our choice of P to form a new simple cycle.
Namely let R1 , R2 be the 2-cells forming the cancellable pair, with R1 ⊂ A and
R2 ⊂ D. Push P across to the other side of R1 . Now R2 can be removed from
D. This is illustrated as the first two configurations in Figure 5, where cancellable
pairs are marked by dots: the first configuration is the case when originally P does
not jump from one side of A to the other around some side of R1 (in which case
the new P crosses A twice at this point); the second configuration occurs when
originally P crosses A along some side of R1 , in which case the new P crosses A
along the other side of R1 afterwards.
In the third case (reduction between A and A), this implies that there is a pair of
2-cells R1 , R2 in L, different from the pair of extremal 2-cells, mapping to the same
This means that we can find a proper subsegment λ of the hypergraph
2-cell of X.
The ladder carrying λ , which has R1 and
segment λ which does not embed in X.
R2 as extremal 2-cells, can now be used to define a smaller quasicollared diagram
as in the rightmost illustration of Figure 5.
Keep reducing cancellable pairs. The only thing to check is that eventually A
is not empty. Observe that reductions between D and D and between D and A
preserve A and λ. So the only way A can become empty is if at some step two
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
4713
Figure 6
C
C
λ′
Figure 7
consecutive 2-cells of A are cancellable. But this means that λ was not immersed,
which it is by definition of a hypergraph segment.
We now extract a collared diagram from a quasicollared one.
Lemma 3.10 (Collaring). If there is a reduced quasicollared diagram F , then there
is a reduced collared diagram F . Moreover, F and F are collared by segments of
the same hypergraph of X.
Proof. Keeping the same notation again, suppose some edge e of P crosses Λ, that
is, consider an edge e in P that is dual to Λ (witnessing for the fact that the diagram
is quasicollared but not collared). Observe that Λ enters D at e. Let λ be the path
of Λ in D issuing from e. As on the left in Figure 6, either λ is simple or λ crosses
itself in D.
If λ crosses itself, then we choose some subpath λ of λ that is a simple loop
in D bounding some topological disc in D, as in the middle diagram of Figure 6.
There is then a diagram D having λ as the hypergraph in its collar. This is
illustrated on the right in Figure 6. (Note that D is only nearly a subdiagram of
D since the map D → D might fail to be injective on ∂D .)
The other possibility is that the path λ is simple in D (Figure 7). In this case,
λ has to exit D by crossing the collar at some 2-cell C, dividing F into two halves.
Pick the half of F that does not contain the corner of F : this provides a new
quasicollared diagram F with C as its corner. (If C happens to be the corner of
F already, then any half will do.)
This new diagram is smaller than F in the sense that the number of intersections between ∂D and the hypergraph segment collaring the diagram decreases. So
repeating the process will eventually provide a collared diagram. The last step is
illustrated in Figure 8.
The new diagram obtained is reduced since F itself is.
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4714
Y. OLLIVIER AND D. T. WISE
C
C
λ′
Figure 8
3.2. Diagrams quasicollared by hypergraphs and paths. We now give a definition of a notion generalizing that of a quasicollared diagram, in which we allow
the collar to consist of segments of several hypergraphs and/or paths in X.
Definition 3.11. Let n 2 be an integer and decompose {1, . . . , n} as a disjoint
union I ∪ J (where I or J may be empty). For i ∈ I, let λi be a hypergraph
segment of length at least 2, carried in the ladder Li . For i ∈ I, also let pi be a
path immersed in Li joining a point in the boundary of the first 2-cell of Li and
a point in the boundary of the last 2-cell of Li . For j ∈ J, let pj be any path
immersed in X.
We suppose that (subscripts mod n):
(1) When i ∈ I and i + 1 ∈ I, then the final 2-cell of Li and the initial 2-cell
Moreover the final edge of λi and the
of Li+1 have the same image in X.
Moreover the (images
initial edge of λi+1 do not have the same image in X.
in X of) initial point of pi+1 and the final point of pi coincide.
of the final point of
(2) When i ∈ I and i + 1 ∈ J, then the image in X
pi coincides with the initial point of pi+1 , and likewise when i ∈ J and
i + 1 ∈ I.
(3) If i ∈ J, then both i + 1 and i − 1 lie in I.
This allows us to define a cyclic path P = pi . Let D be a disc diagram with
boundary path P .
Let A be the disjoint union of Li for i ∈ I and let A be the quotient of A
under the identification of the last 2-cell of Li with the first 2-cell of Li+1 whenever
i, i + 1 ∈ I.
A diagram quasicollared by the λi , i ∈ I and the pj , j ∈ J is the union E =
D pi ,i∈I A.
The corners of E are the initial and final 2-cells of the Li ’s.
It is said to be collared by the λi , i ∈ I and the pj , j ∈ J if E is a genuine disc
diagram.
We say that the hypergraphs λi do not enter E if for i ∈ I, the initial and final
points of λi belong to the boundary of E.
Note that D may be singular, since the map from the unfolded carrier of a
generally identifies a lot of 1-cells and this will result in “spurs”
hypergraph to X
in D. D may even contain no 2-cell in the case that P is a null-homotopic path in
Note also that we do not allow “cornerless” diagrams since we
the 1-skeleton of X.
imposed that two successive hypergraph segments intersect transversely.
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
4715
p2
p1
p3
D
p6
p4
p5
Figure 9. A diagram collared by hypergraphs and paths.
3.3. 2-collared diagrams. A 2-collared diagram is a diagram collared by two hypergraph segments.
The main goal of this section is the following:
Theorem 3.12. Suppose Λ1 and Λ2 are distinct hypergraphs that are embedded
There is more than one point in Λ1 ∩ Λ2 if and only if there exists
trees in X.
a reduced diagram E collared by segments of Λ1 and Λ2 . Moreover, if Λ1 and Λ2
cross at a 2-cell C, then we can choose E so that C is one of its corners.
collared by Λ1 and Λ2 , then the intersecProof. If there exists a diagram E → X
tions of λ1 and λ2 in the two corners 2-cells map to two intersection points in X,
which are distinct since Λ1 and Λ2 are embedded trees.
The converse requires more work, and we outline its proof, which depends upon
lemmas proven in this section.
In Lemma 3.13, we show that if Λ1 and Λ2 intersect twice, then there is a
quasicollared diagram between them.
In Lemma 3.15, we show that if there is a quasicollared diagram between them,
then there is a reduced quasicollared diagram between them.
In Lemma 3.16, we extract a reduced collared diagram between Λ1 and Λ2 , from
a reduced quasicollared diagram.
Lemma 3.13 (Existence of 2-quasicollared diagrams). Suppose there are distinct
intersect
hypergraphs Λ1 and Λ2 which are embedded trees and whose images in X
in more than one point. Then there exists a diagram F quasicollared by Λ1 and Λ2 ;
moreover, its corners can be taken to be an arbitrary pair of distinct 2-cells where
Λ1 and Λ2 intersect.
Proof. Let λ1 , λ2 be hypergraph segments in Λ1 , Λ2 which intersect at the centers
of their first and last edges. Let Li be the ladder carrying λi , and observe that
Let A → X
the first and last 2-cells of L1 , L2 project to the same 2-cells of X.
be obtained by forming the union of L1 and L2 and identifying their first and last
closed 2-cells. Observe that π1 (A) ∼
= Z except for the degenerate case where each
Li consists entirely of these first and last 2-cells.
In this degenerate case, define F = A. Otherwise, let P → A be a simple
be a disc diagram with
closed path representing a generator of π1 (A). Let D → X
boundary path P → X. Let F = A ∪P D.
Lemma 3.14. Let F be a diagram quasicollared by two embedded hypergraph segments λ1 , λ2 . Then there exists a diagram F quasicollared by two subsegments
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4716
Y. OLLIVIER AND D. T. WISE
λ1 , λ2 of λ1 , λ2 , such that the only 2-cells in the intersection of the images of the
are the corners of F . Moreover, F can be chosen to
ladders of λ1 and λ2 in X
contain either corner of F as one of its corners.
Proof. This is more difficult to state than to prove. Let C1 be the first corner of
F . Let C2 be the first 2-cell in the ladder of λ1 , distinct from C1 , which lies in the
Taking the corresponding initial subsegments of λ1
image of the ladder of λ2 in X.
and λ2 and applying Lemma 3.13, we get what we need, preserving corner C1 . Lemma 3.15 (Reducing). Let F be a diagram quasicollared by two hypergraphs
which are embedded trees. Then there exists a reduced diagram F quasicollared by
two segments of the same hypergraphs, and moreover F can be chosen to contain
either corner of F .
Proof. This is similar to Lemma 3.9. Applying Lemma 3.14, we can suppose that
the images of the ladders collaring F intersect only at the two corners of F .
Keep the notation of Definition 3.11. Cancellable pairs between 2-cells both in
D can be removed as usual to lower the total number of 2-cells.
Now for the case of cancellable pairs between A and A. Since each hypergraph
is an embedded tree, the two cells of the pair cannot lie in the same ladder Li . But
the cancellation cannot occur between L1 and L2 either since this would contradict
the conclusion of Lemma 3.14.
Finally, if there is a cancellable pair between D and A, then we can push the
boundary path P across the 2-cell in A (compare Lemma 3.9), obtaining a new
closed path P , which is the boundary path of a disc diagram D with one less
2-cell. (Note that this operation preserves A.)
Lemma 3.16 (Collaring). Let F be a reduced diagram quasicollared by two hypergraphs Λ1 , Λ2 which are embedded trees. Then there is a reduced diagram F collared by Λ1 and Λ2 . Moreover, F can be chosen to share either corner with F .
Proof. Keeping the notation again, note that if the path P does not intersect λ1
and λ2 , then the diagram is collared.
Now suppose that P crosses, say, the segment λ1 . (The argument for λ2 is
identical.) Consider the first such situation on λ1 . Then λ1 can be extended
into a hypergraph segment μ1 that enters D. Since Λ1 is an embedded tree, μ1
cannot cross itself. So μ1 exits D by crossing λ2 , not λ1 . (This contrasts with
Lemma 3.10, where we did not assume that the hypergraph was an embedded
tree.) Then by choosing the part of F lying between μ1 and λ2 we get a diagram
which is quasicollared by λ2 and μ1 , containing the first corner of F . Repeating
the argument with λ2 produces a diagram collared by μ1 and a segment μ2 of Λ2 ,
which ends the proof.
3.4. Diagrams collared by a hypergraph and a path.
Lemma 3.17 (Existence). Let Λ be a hypergraph which is an embedded tree in
Let λ be a segment of Λ. Let γ be an embedded path in X
with the same
X.
endpoints as λ. (Here γ is an edge path which starts and ends at “mid-edge vertices”
corresponding to vertices of Λ.) Then there exists a reduced diagram F quasicollared
by λ and γ.
Moreover, in the case that γ does not intersect Λ anywhere except at its endpoints,
then F is actually collared.
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
4717
L
y
y
1
γ
2
Figure 10
Remark 3.18. γ will be geodesic in our applications. This will serve to study the
metric properties of embedded hypergraphs.
Proof. We proceed as above to get a reduced diagram quasicollared by λ and γ.
Let L be the ladder carrying λ. Let P → L be a simple edge-path with the same
be a disc diagram with boundary path P −1 γ. The
endpoints as λ. Let D → X
2-complex F = D ∪P L is a quasicollared diagram between Λ and γ.
The reduction process is carried out as above. Note that since λ embeds, there
is no pair of cancellable 2-cells between L and L, and so λ is preserved by the
reduction process.
Now suppose that F is not a collared diagram. Then λ passes through an edge
of P . Thus λ can be extended into D by a segment μ in D lying in the same
hypergraph Λ (see Figure 10). Since Λ is an embedded tree, μ cannot cross λ
again. So it has to exit D by crossing γ, thus providing a third intersection point
between Λ and γ.
4. The hypergraphs are embedded trees at d < 1/5
is the Cayley 2-complex associated to a finite presentation of
Henceforth, X
is the
the random group G at density d and length (Definition 1.1), that is, X
universal cover of the standard 2-complex associated to the presentation. In case
so that hypergraphs can be
is odd, we perform a subdivision of all 1-cells of X
defined.
Definition 4.1. Let D be a van Kampen diagram. The external 1-cells of D are
the 1-cells which lie in ∂D. The other 1-cells are internal. A 2-cell of D is external
if its closure contains an external 1-cell; otherwise it is internal.
A pseudoshell of D is a 2-cell R such that |∂R ∩ ∂D| > 12 |∂R|.
A shell of D is a 2-cell R such that the boundary path of D contains a subpath
Q, where Q is a subpath of the boundary path of R, and |Q| > 12 |∂R|.
A spur of D is a 1-cell ending at a valence 1 0-cell on ∂D. Note that D has no
spur if and only if its boundary path is immersed.
The following frequently arising condition is a special case of Greendlinger’s
lemma for C ( 61 ) presentations:
Condition 4.2. For every reduced spurless van Kampen diagram D → X, either
(1) D has at most one 2-cell,
(2) D contains at least two shells.
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4718
Y. OLLIVIER AND D. T. WISE
Theorem 4.3. Let X be the standard 2-complex of some presentation. Suppose
that X satisfies Condition 4.2. Then there is no reduced collared van Kampen
diagram D → X (either cornerless or with a corner).
Consequently, all hypergraphs are trees embedded in X.
Proof. We show that there is no collared diagram. Indeed, suppose there is a
collared diagram. It has no spurs and has more than one 2-cell. But its only
possible shell is its corner. Indeed, every other external 2-shell R contains an edge
of a hypergraph in the interior of D, so any path on ∂R ∩ ∂D has length < 12 |∂R|.
This contradicts Condition 4.2.
Corollary 4.4. For random groups at density d < 1/5, with overwhelming proba
bility, all hypergraphs are trees embedded in X.
Proof. Theorem 6 in [Oll07] states that Condition 4.2 holds with overwhelming
probability for random groups at density d < 1/5. We can therefore apply Theorem 4.3.
The goal of Section 11 is to prove that as soon as d > 1/5, on the contrary there
Figure 21 at the end of
is only one hypergraph, which crosses every 1-cell of X.
the paper shows why hypergraphs are not embedded trees at d > 1/5.
We now turn to the metric aspect of the embeddings.
Theorem 4.5. Consider a random group at density d < 1/5. With overwhelming
1 between two vertices of a hypergraph Λ is at least
probability, the distance in X
(1/2 − 2d − ε) times the minimal number of edges joining them in Λ.
1 between two points y1 , y2 ∈ Λ. It is sufficient to
Proof. Let γ be a geodesic in X
prove the statement of the theorem under the additional hypothesis that γ does
not intersect Λ at any other points.
By Lemma 3.17, there exists a reduced diagram E collared by γ and a ladder L
carrying a segment of Λ.
Since E is collared and not only quasicollared, in particular it is an ordinary
van Kampen diagram. Let n be the number of cells in the ladder L. We have
|∂E| n/2+|γ|. But by Theorem 1.6, up to some ε we have |∂E| (1−2d) |E| (1 − 2d)n and so as claimed we have
1
|γ| n( − 2d).
2
Note that the multiplicative constant does not vanish as d → 15 (compare Figure 18 at d > 15 ). However in the proof of this theorem, we already used that
hypergraphs are embedded trees (a condition needed in our previous study of diagrams collared between a hypergraph and a geodesic).
Corollary 4.6. In random groups at density d < 15 , with overwhelming probability,
the stabilizer of any hypergraph is a free, quasiconvex subgroup.
Proof. Since hypergraphs are trees in the Cayley complex, their stabilizers act freely
on a tree. Since random groups are torsion-free, so are the stabilizers, hence freeness
since torsion-free groups acting freely on trees are free. Now a quasi-isometrically
embedded tree in a hyperbolic space is quasiconvex, since quasi-geodesics remain
at a bounded distance from geodesics.
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
4719
Figure 11
5. 2-collared diagrams
In Section 4, we showed that hypergraphs do not self-intersect at density d < 15 .
It will also be useful to understand the way a pair of hypergraphs can intersect
each other. A naive hope would be that distinct hypergraphs are either disjoint or
intersect in a single point, but this is almost never the case as Figure 11 shows.
However, we will show that at low density, intersecting hypergraphs might “braid”
with each other a bit, but after departing do not converge again, so that intersection
is a relatively local matter.
In Theorem 4.3 we saw that there are no 1-quasicollared diagrams at d < 15 . We
now turn to 2-collared diagrams.
Theorem 5.1. For random groups at d < 16 , with overwhelming probability every
reduced diagram with at least three 2-cells has at least three pseudoshells.
In particular, there exists no reduced 2-collared diagram except the one depicted
in Figure 11.
Corollary 5.2. For random groups at density d < 16 , with overwhelming probability
the following holds: Let Λ1 , Λ2 be two hypergraph rays intersecting in the 2-cell c.
Either they intersect in a 2-cell adjacent to c as in Figure 11, or they do not intersect
anywhere else.
Proof of the corollary. This follows from the theorem by Theorem 3.12.
To prove the theorem, we shall need the following lemma (which will be of
independent use).
Lemma 5.3. Consider a random group at density d < 1/4. With overwhelming
probability, the following holds.
Let D be a reduced spurless van Kampen diagram. Let nps be the number of
pseudoshells in D. Let ni be the number of internal 2-cells of D.
Then the number of external 2-cells in D is at most
(1/2 − d)(nps /2 − ni )
.
1/4 − d
Proof of the lemma. Let A be the set of 2-cells R of D with ∂R ∩ ∂D = ∅. Let
B ⊂ A be those 2-cells R with |∂R ∩ ∂D| > /2 (the pseudoshells of D). Let
C ⊂ B be those 2-cells R with |∂R ∩ ∂D| > (1 − d). Let ne = #A, nps = #B and
nd = #C.
Let D be the diagram obtained from D by removing the 2-cells in C. (D
might not be connected, but this does not matter since Theorem 1.6 applies to
nonconnected diagrams as well.) Let us evaluate the boundary length of D . By
definition, a 2-cell in C contributes at most d edges to ∂D (the ones that were
not on ∂D). All other edges of ∂D were already present on ∂D and belonged to
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4720
Y. OLLIVIER AND D. T. WISE
Figure 12
the boundary of a 2-cell in A − C. A 2-cell in A − B contributes at most /2 edges
and a 2-cell in B − C contributes at most (1 − d) edges. So we have
|∂D | qd + (ne − nps ) + (nps − nd )(1 − d).
2
On the other hand, by Theorem 1.6,
|∂D | (1 − 2d − ε) |D | = (1 − 2d − ε)(ne + ni − nd ),
and the combination of these two inequalities yields the conclusion.
Proof of Theorem 5.1. Suppose there are at most two pseudoshells. By Lemma 5.3,
(1/2−d)(nps /2−ni )
. When nps = 2
the number of external 2-cells is bounded above by
1/4−d
1
and d < 6 , this bound is < 4 and so there are at most 3 external 2-cells. Note that
if ni 1, then there are 0 external 2-cells; hence ni = 0 and there are no internal
2-cells.
So it is enough to rule out diagrams D having exactly three 2-cells r1 , r2 , r3 where
only r1 , r2 are pseudoshells. Since r3 is not a pseudoshell, the internal length of D
is at least /2 and so |∂D| < |D| − 2(/2) = 2. But for d < 1/6, Theorem 1.6
yields |∂D| > 23 |D| = 2 (choosing e.g. ε < (1/6 − d)/10), hence a contradiction.
Note that a 2-collared diagram has at most two pseudoshells (its corners).
Another consequence of the lemma is the following.
Theorem 5.4. For random groups at d < 15 , with overwhelming probability, any
reduced 2-collared diagram has at most five 2-cells and no internal cells.
Proof. For d < 15 and nps = 2, the quantity
nonpositive if ni 1.
(1/2−d)(nps /2−ni )
1/4−d
is less than 6 and is
A less sharp version of this last assertion probably follows from the quasiconvexity obtained in Section 4.
6. Typical carrier 2-cells at d < 1/5
We saw in Theorem 5.1 that at densities less than 1/6, there are no nondegenerate
2-collared diagrams, whereas it is not difficult to check that as soon as d > 1/6,
there are 2-collared diagrams with more than two 2-cells (e.g. the one in Figure 12).
there are no 2-collared
However, as proven in this section, for “most” 2-cells of X,
diagrams having these 2-cells as corner cells.
Let (r1 , . . . , rN ) be the N -tuple of random relators making the presentation,
where by definition N = (2m−1)d . In the sequel we prove that some bad properties
are excluded for the relator r1 with overwhelming probability (these properties are
excluded with high probability for any relator ri with i fixed in advance; however,
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
4721
for any random sample (r1 , . . . , rN ), there might be some i depending on the random
sample, such that ri satisfies these bad properties).
Lemma 6.1. Consider a random group at density d. Then with overwhelming
probability, the following holds.
Let D be a reduced diagram with |D| = 3 and suppose D contains a 2-cell corresponding to the relator r1 . Then the number of internal 1-cells in D is at most
2d + ε.
In particular at d < 1/4 the diagram of Figure 12 does not contain the relator
r1 .
Proof. By Theorem 1.12, the expected number of fulfillings of D is at most
1
ESn (D) (2m − 1) 2 (|∂D|−(1−2d)|D|) = (2m − 1)d|D|−L ,
where L = 12 ( |D| − |∂D|) is the internal length of D.
By the symmetry of all relators in the presentation, the expected number of fulfillings of D having the fixed relator r1 as one of its 2-cells is at most
3(2m − 1)−d ESn (D) (the 3 accounts for the choice of the 2-cell in D we are talking
about). Thus the probability that there exists such a fulfilling is at most
3(2m − 1)−d ESn (D) 3(2m − 1)2d−L ,
which decreases exponentially fast if L 2d + ε.
mapping
Lemma 6.2. Consider a random group at d < 1/4. Let c be a 2-cell in X
to r1 . Let Λ1 be any hypergraph through c. Then, with overwhelming probability,
there exists a hypergraph Λ2 through c which is locally transverse to Λ1 in the
following sense:
If Λ1 (resp. Λ2 ) intersects ∂c at the points xl , xr (resp. xt , xb ), then any 2-cell
adjacent to c contains at most one of xl , xr , xt , xb . In particular, Λ1 , Λ2 do not
form a 2-collared diagram as illustrated in Figure 11.
Moreover, there are at least (1/2 − 2d − ε) choices for Λ2 .
Proof. There are two paths ptop , pbot from xl to xr in ∂c. (Note that since a random
word of large length very probably contains every generator of the group at least
is contained in several 2-cells.) Now define ctl (“topleft”) as
once, every 1-cell in X
any 2-cell adjacent to c and containing xl so that the length of the intersection of
∂ctl with ptop is maximal. (Note that thanks to Corollary 1.11, the intersection of
∂ctl with ∂c is connected.) Let tl be this length.
Define ctr , cbl , cbr and tr , bl , br similarly (see Figure 13). We may have ctl = cbl
and ctr = cbr , but this does not affect our argument.
We have tl < 2d − bl + ε; otherwise the diagram c ∪ ctl ∪ cbl would contradict
Lemma 6.1. (In order to get a genuine van Kampen diagram in case ctl ∩ctb contains
some 1-cells, we have to unglue a bit ctl below xl and cbl above xl . This is consistent
with our definition of tl and bl as the length of the intersection with resp. ptop
and pbot .)
Similarly, we have tl 2d−tr +ε, br 2d−bl +ε, and br 2d−tr +ε.
Set L1 = max(tl , br ) and L2 = max(bl , tr ). We have L1 2d + ε − L2 .
Since d < 1/4 we can choose ε so that 2d + ε < 1/2, and so L1 < /2 − L2 (and the
discrepancy is at least (1/2 − 2d − ε)).
Now take any
point xt on ptop so that the distance from xt to xl lies in the interval
L1 , (/2 − L2 ) . There are at least (1/2 − 2d − ε) such points. Let xb ∈ pbot be the
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4722
Y. OLLIVIER AND D. T. WISE
ctl
ctr
cbl
cbr
Λ2
Figure 13
opposite point in c. By construction, xt and xb do not lie in any of ctl , ctr , cbl , cbr .
By maximality of these latter 2-cells among 2-cells adjacent to c containing either
xl or xp , no other 2-cell adjacent to c can contain two of the x’s.
Now let of course Λ2 be the hypergraph through xt and xb .
Lemma 6.3. Consider a random group at d < 1/5. With overwhelming probability,
there is no reduced 2-collared diagram admitting r1 on one of its corner cells, except
the one in Figure 11.
Proof. Let D be a 2-collared diagram having relator r1 as one of its corner cells. By
Theorem 5.4, we only have a finite number of diagrams to check. We can thus obtain
overwhelming probability by intersecting finitely many events with overwhelming
probability.
First, suppose that the other corner 2-cell of D has less than (1 − d) edges on
the boundary of D. Since in a 2-collared diagram, every 2-cell except maybe the
corners has less than half its length on the boundary, this means that we have
|∂D| + (1 − d) + (|D| − 2)/2. So the expected number of fulfillings of this
diagram is, by Theorem 1.12, at most
1
ESn (D) (2m − 1) 2 (|∂D|−(1−2d)|D|) (2m − 1)(1/2−d/2+|D|(d−1/4)) .
By symmetry of all (2m − 1)d relators in the presentation, the expected number
of fulfillings of D having the fixed relator r1 as its corner 2-cell is at most (2m −
1)−d ESn (D), and so the probability that there exists such a fulfilling is at most
(2m − 1)−d ESn (D) (2m − 1)(1/2−3d/2+|D|(d−1/4))
so that if
1/2 − 3d/2 + |D| (d − 1/4) < 0,
then this probability is exponentially small. So if |D| > 1/2−3d/2
1/4−d , then with overwhelming probability this does not happen. For d < 1/5 the right-hand side is less
than 4. So the only possibility is the three 2-cell diagram depicted in Figure 12.
But we have just excluded it in Lemma 6.1.
Second, suppose that the other corner of D has more than (1 − d) on the
boundary. Then we get the same conclusion by reasoning on the new diagram D
obtained by removing this corner.
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
4723
7. Codimension-1 subgroups at d < 1/5
is an arbitrary simply connected 2-complex, each 2-cell of
Here for some time X
is equipped with its hypergraph system as in
which has even boundary length; X
is the Cayley 2-complex
Definition 2.1. We shall then return to the case when X
of a presentation of a random group at density d and length (up to subdivision
of the 1-cells if is odd).
the orientation-preserving
Definition 7.1. For a hypergraph Λ in the 2-complex X,
+
stabilizer Stabilizer (Λ) of Λ is the index 2 subgroup of Stabilizer(Λ) that also
− Λ. Equivalently,
stabilizes each of the two halfspaces which are components of X
+
+
+
−Λ.
Stabilizer (Λ) equals Stabilizer(H ), where H is one of the components of X
We now prove the existence of codimension-1 subgroups at density d < 1/5.
These subgroups are orientation-preserving stabilizers of hypergraphs passing
through “typical” 2-cells of X.
Lemma 7.2 (Codimension-1 criterion). Suppose that the discrete group G acts
and that the system of hypercocompactly on the simply connected 2-complex X
graphs in X is locally finite and cocompact (meaning that the hypergraphs in X/G
are compact and there is only a finite number of them). Suppose that two distinct
hypergraphs Λ1 and Λ2 cross each other at a single point. Suppose that each Λi is
an embedded tree with no leaves.
Then Hi = Stabilizer+ (Λi ) is a subgroup of G with a relative number of ends
e(G, Hi ) = 2.
We have to prove that X̄1 has
Proof. Let H1 = Stabilizer+ (Λ1 ). Let X̄1 = H1 \X.
at least two ends.
into two connected components. Let
According to Lemma 2.3, Λ1 separates X
Λ̄i be the image of Λi in X̄1 . Suppose a component of X̄1 − Λ̄1 is compact. Consider
the edge e of Λ2 , where Λ2 crosses Λ1 . Using that Λ2 is a leafless tree, extend e
to a ray r in the direction of the halfspace mapping to the compact component of
X̄1 − Λ̄1 .
By compactness, the projection r̄ of r to X̄1 must pass through Λ̄1 a second
time. Indeed, let u be the first combinatorial subpath of the graph r whose initial
and final vertices have the same projection in X̄1 , so that the projection ū is a
cycle in X̄1 ; such a path necessarily exists by compactness. Thus r = suv, and s
is minimal with this property. Consider the path p = s̄ūs̄−1 . We show that p is
an immersed path. Indeed, if ūs̄−1 has a backtrack, then ū = w̄ē and s̄ = s̄ ē. Let
ū = ēw̄. Then ū is a closed path with |s̄ | < |s̄|, so ū occurs earlier than ū, which
contradicts the choice of u.
is a segment of Λ2 , which is not closed since it is a subpath
The lift p̃ of p to X
of r which is a geodesic in Λ2 .
Finally, let q1 be a path in Λ1 which projects to a path in Λ̄1 with the same
two
endpoints as p. The common endpoints of p and q1 provide, after lifting to X,
intersections of the hypergraphs in X, which contradicts the assumption.
Lemma 7.3. With overwhelming probability, at any density, the first relator r1 in
the random presentation involves all generators.
Consequently, hypergraphs have no leaves, and any hypergraph passes through a
2-cell bearing relator r1 .
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4724
Y. OLLIVIER AND D. T. WISE
Proof. The first assertion is a consequence of the law of large numbers. It follows
that any 1-cell of the Cayley 2-complex of a random group is contained in a 2-cell
bearing relator r1 ; hence, hypergraphs are leafless.
Theorem 7.4. With overwhelming probability, random groups G at density d < 15
have a subgroup H which is free, quasiconvex and such that the relative number of
ends e(G, H) is at least 2.
This subgroup can be taken to be the orientation-preserving stabilizer of any
hypergraph.
Proof. Let r1 be the first relator in the presentation. Let Λ1 be a hypergraph.
By Lemma 7.3, this hypergraph travels through a 2-cell c bearing r1 . Let Λ2 be
the hypergraph provided by Lemma 6.2. By Theorem 4.3, these hypergraphs are
embedded trees, leafless by Lemma 7.3.
By Lemma 6.3 (in conjunction with Lemma 6.2), Λ1 and Λ2 do not form any reduced collared diagram with corner 2-cell c, and so by Theorem 3.12, they intersect
only at c.
Now apply Lemma 7.2 to get the number of relative ends. The other assertions
follow from Corollary 4.6.
Corollary 7.5. Suppose that d < 1/5. Then with overwhelming probability, a
random group does not have Property (T ).
Proof. It was shown in [NR98] that groups having a subgroup with more than one
relative end do not have Property (T ).
8. Carriers are convex at d < 1/6
Recall that the carrier of a hypergraph is the set of 2-cells the hypergraph passes
is convex if for any two
through. We say that a subcomplex Y of the 2-complex X
is included in Y .
0-cells in Y , the shortest path between them in X
Theorem 8.1. The following holds with overwhelming probability at d <
each hypergraph Λ its carrier Y is a convex subcomplex of X.
1
6:
For
Proof. Let y1 , y2 be two points on Y (which may not lie on the same side of Λ).
joining y1 to y2 . We want to show that γ lies in Y .
Let γ be a geodesic in X
Suppose that γ does not lie in Y . We can decompose γ into subparts which
either are included in Y , or intersect Y only at their endpoints. There is nothing
to prove in the former case, so we can suppose that the intersection of γ with Y is
exactly {y1 , y2 } (and in particular y1 and y2 lie on the same side of Λ).
Let L be a ladder in Y between 2-cells containing y1 and y2 , and let y1 and
y2 be the extremal points of the hypergraph segment contained in L. Let γ =
[y1 y1 ].γ.[y2 y2 ] be the union of γ with paths joining yi to yi , respectively. Let D be
a reduced van Kampen diagram collared by L and γ , as provided by Lemma 3.17
(see Figure 14). According to this lemma, D is collared and not only quasicollared
since γ does not intersect the hypergraph except at its endpoints.
Thanks to the collaring, every 2-cell of L except the two extremal ones has less
than half its length on the boundary of D.
Now let c be a 2-cell lying on the boundary of D but not belonging to L (so that
∂c ∩ ∂D ⊂ γ). Since γ is a geodesic, this means that the length of ∂c ∩ γ is no more
than half the boundary length of c (otherwise we could shorten the geodesic).
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
4725
L
y1′
y1
y2
γ
y2′
Figure 14
d/2
2d
d/2
d/2 d/2
(1 − 3d)
Figure 15. The carrier is not convex at d > 1/6.
So every 2-cell of D except maybe the two extremal cells of L has no more than
half its length on the boundary of D, so that D has at most two pseudoshells. But
at d < 1/6 this is ruled out by Theorem 5.1, except when D = L has only two
2-cells, which was to be proven.
Remark 8.2. At density d < 1/6, if a 2-cell R is such that ∂R is included in the
carrier Y of some hypergraph Λ, then R itself is included in Y . Indeed, otherwise
any hypergraph through R would meet Λ twice, contradicting Corollary 5.2.
Remark 8.3. It is not difficult to see that carriers are not convex when d > 1/6.
Indeed, at density > d, with overwhelming probability there exists a diagram as
depicted in Figure 15, in which the bottom 2-cell has more than half its boundary
length on the boundary of the carrier, thus making it shorter to turn around from
below. To see that such a diagram exists, first select the two middle relators of the
diagram: they have to share a length 2d, and at density > d such a pair of relators
exists with overwhelming probability by Proposition 10 in [Oll05b]; then one has
to find the right and left relators, each of which has to share a length d with the
rest of the diagram, and at density > d this is possible thanks to Proposition 9 in
[Oll05b].
Remark 8.4. At density d < 14 , any 2-cell R is convex. Indeed, let p be an immersed
path in ∂R with |p| /2, and let p be a distinct immersed path with the same
endpoints as p and |p | |p|. Then p p−1 is the boundary of a van Kampen diagram
D, which thus satisfies |∂D| . But according to Theorem 1.6 at d < 1/4, this
implies that D has at most one 2-cell, which easily implies that D = R and |p| = /2.
9. Separation by hypergraphs at d < 1/6
the number of
The goal of this section is to prove that for any two points in X,
hypergraphs separating them grows linearly with their distance. This will enable
us to apply a CAT(0) criterion in Section 10.
For p, q ∈ X we let #(p, q) equal the number of hypergraphs Λ such that p and
q lie in distinct components of X − Λ.
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4726
Y. OLLIVIER AND D. T. WISE
Theorem 9.1. The following holds with overwhelming probability at d < 16 : For
0 we have
all p, q ∈ X
1 1
− d − ε (d(p, q) − 6) .
#(p, q) 2 6
Corollary 9.2. A random group at density d < 1/6 has the Haagerup property.
Proof of the corollary. A discrete group acts properly on its Cayley 2-complex
(equipped, say, with the edge metric on the 1-skeleton and Euclidean metrics on
each 2-cell). Now, since the hypergraphs are embedded trees, the system of hypergraph turns this 2-complex into a space with walls [HP98], and the theorem
above states that the wall metric is equivalent to the edge metric. So the group
acts properly on a space with walls, which by a folklore remark (see, e.g., [CMV04])
implies the Haagerup property.
For the proof of Theorem 9.1 we will need two lemmas.
Lemma 9.3. The following holds with overwhelming probability at d < 16 : For each
2-cell R, any two disjoint pieces P1 , P2 in ∂R satisfy |P1 | + |P2 | < 3d + ε < 2 .
Proof. This follows directly from Theorem 1.6: indeed, we can form a three-2-cell
diagram involving the two pieces, and at d < 1/6 its internal length is less than
/2.
Lemma 9.4. The following holds with overwhelming probability at d < 16 : Let Λ
be a hypergraph passing through a 2-cell R. Then there exists another hypergraph
Λd passing through R, such that Λ ∩ Λd consists of a single point.
Actually there are at least (1/2 − 3d − ε) choices for Λd .
Sketch following Lemma 6.2. The proof is identical to Lemma 6.2, except that at
density < 1/6 we do not have to fix the relator in advance (and we use Lemma 9.3
in place of Lemma 6.1). See Figure 13.
Proof of Theorem 9.1. Let γ be a geodesic between p and q. We show that for each
length-3 subpath σ of γ, either the hypergraph through an edge at the middle of σ
crosses γ only once, or there are at least (1/2 − 3d − ε) edges in σ, the hypergraph
through which crosses γ only once. This shows that whenever d(p, q) 6 the number of choices for such a hypergraph is at least (1/2 − 3d − ε)(6)−1 (d(p, q) − 6).
Let e1 be an edge at the middle of σ. Let Λe be the corresponding hypergraph.
If Λe ∩ γ = Λe ∩ e1 , then we are done. Otherwise Λe crosses γ at a first other edge
e3 . Without loss of generality, assume that e1 < e3 in the ordering on γ.
By convexity, the subpath [e1 e3 ] of γ lies in the carrier of Λe . Also remember
are convex.
that, by Remark 8.4, all 2-cells of X
First observe that e1 and e3 cannot be consecutive dual 1-cells of Λe crossing
the same 2-cell R, for then |[e1 e3 ]| = 2 + 1, but the complementary part of ∂R has
length 2 − 1, so γ would fail to be a geodesic.
In the other extreme, if there is more than one dual 1-cell of Λe between e1 and
e3 , then we let R be the second 2-cell in the ladder of Λe between e1 and e3 . We
then apply Lemma 9.4 to obtain a hypergraph Λk passing through R that intersects
Λe at a single point (and we have (1/2 − 3d − ε) choices for Λk ).
This hypergraph Λk crosses γ in a single edge k. Indeed, suppose Λk crossed γ
in a second edge k2 . If e1 < k2 < e3 , then Λk crosses Λe in a second point, which is
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
f3
e1
g3
e2
e
4727
e3
Re
R
g2
g
f2
Rf
f1
f
Figure 16
impossible. Similarly, if k2 < e1 , then by convexity [k2 k] ⊂ Yk and hence e1 ⊂ Yk .
Consequently, Λe crosses Λk at the center of a 2-cell containing e1 on its boundary.
By hypothesis, this 2-cell cannot be R, and so Λe and Λk intersect in more than one
point, which is impossible. An analogous argument excludes the possibility that
e3 < k2 .
Finally, we consider the case where e1 and e3 are separated by a single dual 1-cell
e2 . Let Re be the 2-cell between e1 and e2 , and let R be the 2-cell between e2 and
e3 (see Figure 16).
Applying Lemma 9.4, let Λf be a hypergraph passing through R that intersects
Λe at a single point. Let Yf be the carrier of Λf .
If Λf intersects γ in a single 1-cell f3 , then we are done (and there are (1/2 −
3d − ε) choices for Λf ). Otherwise, let f1 be the next 1-cell in γ that Λf passes
through. As in the first case above, since γ is a geodesic, f3 and f1 cannot be
consecutive dual 1-cells of Λf . We may also assume that there is no more than one
dual 1-cell f2 between them, for otherwise, as in the construction of Λk considered
above, we could find a third hypergraph Λh intersecting Λf at a single point which
is at the center of a 2-cell separating f1 and f3 and we would be done (once more
with at least (1/2 − 3d − ε) choices for Λh ).
Finally, we consider the case where there is exactly one dual 1-cell f2 between
f3 and f1 in Yf . We refer the reader to Figure 16.
Let Rf be the 2-cell in Yf between f2 and f1 . Let Pe = R ∩ Re and let Pf =
R ∩ Rf . Note that Pe and Pf are disjoint. Indeed, otherwise the complement of
γ ∩ R is the concatenation of two pieces and this has length less than 3d + ε < 2
by Lemma 9.3, and so γ would not be a geodesic. Consequently, Pe and Pf are
separated on either side of R by R ∩ γ and an edge g2 . More precisely, since
|γ ∩ R| /2 and |Pe | + |Pf | < 3d + ε, we have at least (1/2 − 3d − ε) choices
for g2 .
Let Λg be the hypergraph dual to g2 , with carrier Yg , and let g3 be the 1-cell in
∂R that is antipodal to g2 . Note that since g2 lies between e2 and f2 , the 1-cell g3
lies between f3 and e3 in ∂R, and hence Λg crosses γ in g3 . We will show that Λg
does not cross any other edge of γ.
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4728
Y. OLLIVIER AND D. T. WISE
Note that g2 is not a 1-cell of γ, since as γ is a geodesic it cannot contain two
opposite 1-cells of a 2-cell.
By definition, g2 does not lie in Re ∩ R. Neither does g3 , since f3 is between e2
and g3 on ∂R and by definition Λf does not pass through Re . So by Corollary 5.2,
Λg does not pass through Re , i.e. Re ⊂ Yg .
Let a be the last 1-cell in γ before R. Suppose that Λg crossed a 1-cell h of
γ with h a. Since Re does not lie in Yg we have h e1 . But then since e1
lies in the geodesic γ between g3 and h, by convexity of Yg we get e1 ⊂ Yg . Since
e2 ⊂ Yg too by construction, by convexity and Remark 8.2 we get Re ⊂ Yg , which
is a contradiction.
Let b be the first 1-cell in γ after R. A similar argument shows that Λg cannot
cross a 1-cell h of γ with b h.
Note that we used only three 2-cells in the construction, so that the intersection
of γ with these has length at most 3/2; hence all hypergraphs considered are dual
to some edge inside the subpath σ. We have to consider a segment σ of length 3
because we do not know on which side of e1 the edge e3 will fall.
10. CAT(0) cubulation at d < 1/6
We now proceed to the geometrization theorem at d < 1/6. We begin by listing
the following criteria from [HW04]:
be a 2–complex equipped with a collection
Theorem 10.1 (Local Finiteness). Let X
of hypergraphs satisfying the following properties.
is locally finite.
(1) X
(2) The hypergraph system is uniformly locally finite.
(3) There is a constant K so that for each n ≥ 1, every pair of points at a
distance at least nK apart are separated by at least n distinct hypergraphs.
(4) There is a constant δ such that every hypergraph triangle is δ–thin.
is locally finite.
Then the cube complex C associated to X
be locally finite with a locally finite cube comTheorem 10.2 (Properness). Let X
then the induced action of Γ on C
plex C. If Γ acts properly discontinuously on X,
is also properly discontinuous.
The following is formulated in [HW04], but was first proven by Sageev in [Sag97].
Theorem 10.3 (Cocompactness). Suppose Γ acts properly and cocompactly on X.
Then Γ acts properly and cocompactly on C provided that the following conditions
hold.
is δ–hyperbolic.
(1) X
(2) The hypergraphs are quasiconvex.
(3) The hypergraph system is locally finite.
We can now prove our second main theorem:
Theorem 10.4. With overwhelming probability, a random group at density d <
acts freely and cocompactly on a CAT(0) cube complex C.
1
6
Proof. As shown by Gromov [Gro93] (see also [Oll04]), G is hyperbolic and torsionfree with overwhelming probability at d < 12 .
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
4729
Figure 17. At d > 1/4, the hypergraph is trivially 1-dense in X.
The quasiconvexity of the hypergraphs, and hence the codimension-1 subgroups
that are their orientation-preserving stabilizers, was proven in Theorem 4.5 at d < 15
with overwhelming probability. In fact, the convexity of hypergraph carriers was
proven in Theorem 8.1 at d < 16 with overwhelming probability.
and the system of hypergraphs is obvious in
The uniform local finiteness of X
our case.
Applying Theorem 10.3, we see that G acts cocompactly on the cube complex
C.
Since G is hyperbolic, and the hypergraphs embed by quasi-isometries, we see
are δ-thin for some δ depending on the hyperthat all hypergraph triangles in X
bolicity constant for G and the quasi-isometry constants for the hypergraphs.
Finally the linear separation condition was proven in Theorem 9.1.
Thus the cube complex is locally finite by Theorem 10.1. Consequently G acts
properly discontinuously on C by Theorem 10.2.
Since G is torsion-free we see that the action is free, and we are done.
The crucial difference between the 1/5 and 1/6 cases was the separation of any
two points by a linear number of hypergraphs proven in Theorem 9.1. We suspect
that this should hold at density d < 1/5, but adapting the proof of Theorem 9.1
to this case involves the analysis of many particular cases corresponding to the
existence of small 2-collared diagrams at density 1/6 < d < 1/5.
Conjecture 10.5. With overwhelming probability, random groups at density d <
act freely and cocompactly on a CAT(0) cube complex.
1
5
11. The unique hypergraph is π1 -surjective at d > 1/5
The next theorem shows that our approach fails at density d > 15 .
Theorem 11.1. Let Λ be a hypergraph in the standard 2-complex X = X/G
of a
1
∼
random group presentation at density d > 5 . Then π1 Λ → π1 X = G is surjective.
if the hypergraph Λ contains the midpoint
Proof. It is equivalent to prove that in X,
of some edge e1 , then it also contains the midpoint of any other edge e2 sharing a
vertex with e1 . In particular it follows that there is only one hypergraph in X.
First, let us give a very simple argument proving this under the stronger condition
that d > 1/4. Indeed, in this case, with overwhelming probability there are two
relators sharing a piece of length /2. Thus the diagram of Figure 17 occurs with
overwhelming probability (and then, since all reduced words have equal probability,
every combination of generators of the group will occur on edges e1 and e2 ).
We will show that as soon as d > 1/5, with overwhelming probability there exists
a van Kampen diagram as depicted in Figure 18, where the edges e1 and e2 are
the rightmost edges of the diagram. Actually, we will prove that at d > 1/5, very
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4730
Y. OLLIVIER AND D. T. WISE
Figure 18. At d > 1/5, the hypergraph is 1-dense in X.
Figure 19
Figure 20
r2
/10
2/5
/5
w
/10
r1
r3
Figure 21. Hypergraphs do not embed at d > 1/5.
probably a van Kampen diagram as depicted in Figure 19 occurs; indeed, as the
latter diagram has one more internal edge, it has smaller probability to be realized
by random relators; hence if the latter diagram occurs with high probability, then
so does the former (and then, since all reduced words have equal probability, every
combination of generators of the group will occur on edges e1 and e2 ).
The diagram in Figure 19 can be obtained by gluing two copies of the diagram
in Figure 20. Let us divide the set of relators R of the random groups into two
(arbitrary) halves R1 and R2 each with 12 (2m − 1)d relators. We will, at first,
consider diagrams with relators only in R1 or only in R2 .
Now fix an integer K 3 and let DK be the diagram as in Figure 20 with K
2-cells, and with the additional constraint that the “vertical” sides of the diagram
have length /5.
For K = 3 we get the diagram in Figure 21. In [Oll07] (last section), it is
established that such a diagram occurs with overwhelming probability as soon as
d > 1/5. This immediately proves that hypergraphs do not embed at d > 1/5, so
that Corollary 4.4 is sharp. Note that d = 1/5 is the smallest value for which the
topology of Figure 21 will yield a nonembedded hypergraph, in view of Theorem 1.6;
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CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6
4731
note also that the results in [Oll07] are unrelated to hypergraphs but d = 1/5
already plays a special role.
1/5
the diagram
We will, by induction, establish that for any K < K0 = 3 + d−1/5
DK occurs with overwhelming probability. More precisely, for a random group
at d > 1/5 consider the set of all van Kampen diagrams with shape DK occurring in the group and containing only relators from R1 , and let NK be the number of distinct reduced words of length /5 appearing on the rightmost side of
all these diagrams. We are going to show that, with overwhelming probability,
1
(2m − 1)(K−3)(d−1/5) . We know that, with overwhelming probability,
NK 8K−3
N3 1.
We will use the following elementary lemma, the proof of which is omitted.
Lemma 11.2. Let w be a random reduced word of length . Let w1 and w2 be
disjoint subwords of w which are separated by at least one letter of w. Then the law
of w2 knowing w1 is close to the law of a random reduced word independent from
w1 .
Here “close to” means that the ratio of probabilities lies between 1 − c and 1 + c
for some small universal constant c < 1; moreover, the error decreases exponentially
with the number of letters separating w1 and w2 .
Now, for K 4, the diagram DK is obtained from DK−1 by gluing a relator
along a subword of length /5 on the rightmost side. If the subword on the rightmost
side of DK−1 is fixed, a random relator has probability (2m − 1)−/5 to fulfill this
condition. Since there are by definition NK−1 possibilities for the subword on the
rightmost side of DK−1 , a random relator has probability NK−1 (2m − 1)−/5 to
form the rightmost 2-cell of a diagram with shape DK .
Since there are 12 (2m − 1)d random relators in R1 , on average 12 NK−1 (2m −
(d−1/5)
distinct diagrams DK occur. This is only an average, but standard
1)
(e.g. Chernoff-type) deviation results show that with overwhelming probability, at
least 14 NK−1 (2m − 1)(d−1/5) distinct diagrams DK occur. To compute NK we
must consider the set of all subwords of length /5 on the rightmost side of all
these diagrams. According to the lemma above we get 14 NK−1 (2m − 1)(d−1/5)
independent reduced words with (almost) uniform distribution. All these random
words are not necessarily distinct (there might be some repetitions), but for K <
K0 this number is negligible compared to the total number of reduced words so
that repetitions are negligible and, with overwhelming probability, we get at least
1
(d−1/5)
distinct reduced words. So NK 18 NK−1 (2m − 1)(d−1/5)
8 NK−1 (2m − 1)
as needed.
Choose K = K0 − 1. We know that NK 8K10 −4 (2m − 1)(K0 −4)(d−1/5) (2m − 1)/10 (if 1/5 < d 1/4). So we can produce more than (2m − 1)/10 van
Kampen diagrams with shape DK , with more than (2m − 1)/10 distinct random
reduced words of length /5 on their rightmost side. For this we have been using
only the relators in R1 ; using the other half of the relators in the presentation, we
get a second set of diagrams with the same characteristics, distinct (as van Kampen
diagrams) from the first set. Since there are (2m − 1)/5 reduced words of length
/5, by the probabilistic pigeon-hole principle (e.g. [Oll05b], p. 31) two random
sets of more than (2m − 1)/10 reduced words of length /5 will have a nontrivial
intersection with overwhelming probability. Thus, we can find two distinct copies of
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4732
Y. OLLIVIER AND D. T. WISE
the diagram in Figure 20, sharing the same rightmost subword. These two diagrams
can be joined to form the diagram in Figure 19. This ends the proof.
Acknowledgments
We would like to thank the anonymous referee for helpful suggestions and a
careful reading of our paper.
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´
CNRS, UMPA, Ecole
normale supérieure de Lyon, 46, allée d’Italie, 69364 Lyon cedex
7, France
E-mail address: [email protected]
Current address: CNRS, LRI, Université Paris-Sud, Bat. 490, 91405 Orsay cedex, France
E-mail address: [email protected]
Department of Mathematics, McGill University, Montreal, Québec, Canada H3A
2K6
E-mail address: [email protected]
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